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I study an economy with two goods ($x$ and $y$) and two consumers ($A$ and $B$). First there is no trade between $A$ and $B$, so they consume their endowments $e^A_x$ and $e^B_x$ and reach utility levels $u^A_1$ and $u^B_1$. Thereafter, trade develops between $A$ and $B$ and they reach utility levels $u^A_2$ and $u^B_2$.

I now want to find how much better off each consumer has become due to trade. I understand that looking at the increase in utility does not give any meaning, since the utility function is an ordinal function. From what I can gather, the correct way of doing this is to calculate the compensating variation $CV$ - how much more of the nominal good they need in autarchy in order to reach $u^A_2$ and $u^B_2$, respectively.

However, whouldn't comparing the value of each consumer's endowments (i.e. their autarchy consumption) and the value of the goods they end up consuming after trade also tell us how much better off they are after trade?

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It wouldn't. The value of consumer's endowments and the value after they trade should be the same. i.e. Neither consumer becomes richer in terms of money. They just allocate their money towards buying goods they value more.

Additionally, prices are a function of both consumers' demands while CV only depends on the demands of one consumer. Consider the following example

In autarky consumer 1 has endowment, $e_1=(1,1)$ and utility $$u_1(x,y) = x + 2y $$ so $u_1(1,1)=3$ Now there is trade where consumer 2 has endowment, $e_2=(1,1)$ and utility $$u_2(x,y) = 2x + y $$ When the consumers trade the equilibrium price is $p_x = p_y$ and equilibrium consumption bundles are that consumer 1 gets $(0,2)$ and consumer 2 gets $(2,0)$. The value of both consumers' endowments and equilibrium allocations is $2p_x$. But in terms of compensating variation (with $x$ as the nominal good) consumer 1 would require 2 units of $x$ to move from free trade to autarky while consumer 2 would require 1 unit of $x$.

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